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Paper ST-156
Optimizing the Marketing Mix
Patralekha Bhattacharya, Thinkalytics
to the most efficient channels. Various marketing
models have therefore been developed to help
calculate the effectiveness (dollar sales per GRP) as
well as the return on investment on advertising
expenditures (dollar Sales per unit of expenditure)
and provide direction and accountability to marketing
spending. These models help to determine how much
a business should spend on advertising in order to
increase sales and get a positive return on their
investment. The analytical and statistical methods
that are used to quantify the effects of media and
marketing efforts on a product's performance is
referred to as “Marketing Mix Modeling”.
1. ABSTRACT
Practitioners in the media industry are frequently
faced with the problem of diverting limited resources
among several alternative channels. The paper looks
at this problem of optimal allocation of limited
resources to maximize an objective function within
the context of media mix modeling. If the objective
function is linear or convex these problems can be
easily solved by applying standard methods of linear
or geometric programming. However, in the media
world, non linear relationships between sales and
media variables and advertising carry over effects
can lead to non convex objective functions that make
matters more complicated.
In order to reach various different audiences, media
planners often have to allocate their resources
between various media including Television, print,
magazine, radio, internet, out of home etc. They may
also distribute their advertising budget between
different products. For instance, in the entertainment
industry, the advertisement budget for a movie may
be spent on the Theatrical version of the movie as
well as on the DVD version. Therefore, an equally
important topic of interest to marketing mix modeling
professionals is how best to allocate a fixed budget
among various different media instruments(or
products) in order to maximize the incremental
returns from that investment. The optimal portfolio
finds the relative levels of the two or more media
vehicles (or products, channels etc) that maximize
retail sales while keeping a fixed budget. For example
we might be required to find the mix between :15
and :30 second television spots that maximize retail
sales at a given level of spending. Another problem
may be to find the mix between spending on
Theatrical media and spending on DVD media that
maximizes the Total Revenue (Box office sales +
DVD sales). Alternatively, we may be required to
obtain the optimal allocation of spending between the
various media types such as TV, Magazine,
Newspaper and Online.
In this paper, we develop an alternative solution using
an algorithm (which we deploy with the help of a SAS®
macro) that can be used to obtain a “close to” global
solution for the optimal media mix. The algorithm
uses an iteration method to obtain the value of the
objective function for all potential allocations. The
allocation that maximizes the objective function can
then be chosen as the most favorable media mix.
Since the method works with the transformed
variables, non linear transformations and complicated
carryover effects can be incorporated within this
structure. This allows for a more realistic estimation of
media impacts.
2. INTRODUCTION
Most businesses spend millions of dollars on
advertising every year. According to data released by
TNS Media Intelligence, total advertising expenditures
in 2006 were $149.6 billion, an increase of 4.1%
since 2005. The 2007 advertising market slowed
down somewhat due to pessimism about general
economic conditions; in spite of that, advertising
expenditures in 2007 amounted to $148.99 billion, up
0.2% compared to 2006. In recent years, National TV
has accounted for the majority (about 30%) of the
advertisement budget, with magazines close behind
at about 20%.
3. SHORTCOMINGS OF EXISTING
METHODS
Although advertising expenditures have increased in
recent years, marketers are increasingly cautious
about the dollars spent on advertising and demand
more accountability from their communication
partners about their marketing spending, making it
imperative for media planners to invest their
marketing dollars in the best possible manner to
maximize returns in terms of incremental sales.
Seeking to justify their marketing spending, marketing
planners in turn have solicited the help of statisticians
and econometricians to quantify the effects of
marketing spending as well as to direct the spending
The problem of finding an optimal mix has received a
good deal of attention in the marketing literature.
Doyle and Sanders (1990) proposed a multiproduct
advertising budgeting model which allowed product
categories to have own and cross effects. This model
can easily be extended to address the problem of
allocation of a fixed budget between different media
vehicles. However, most of these studies have
focused on demand curves that are either linear or
linear additive and have used demand functions
which can be easily solved using algebraic methods
1
for the optimal solution. Few studies that have used
non linear demand and cost functions still assume
that the objective function is convex or can be at least
transformed into a convex functional.
may not make business sense. Looking at several
alternative solutions which make more business
sense may help to choose the best marketing mix.
In practice however, the media industry uses various
transformations some of which may be non-linear and
difficult to manipulate algebraically. For instance,
practitioners in the media world often assume that the
media response function is S-shaped, i.e. has an
initial convex and subsequently a concave section.
The theoretical reason behind this shape is that a socalled threshold effect takes place, i.e. the
phenomena that marketing efforts are not effective
until they exceed a certain minimum level (Hanssens
et al. 2001, p. 113). Most managers and practitioners
agree that advertising threshold effects exist so that
there are levels of advertising below which there is
essentially no sales response (Corkindale and
Newall, 1978 ; Ambler ,1996). This has led various
researchers to recognize an S shaped advertising
response curve. When two or more media variables
have such non-linear S shaped response curves the
problem of choosing the optimal media mix among
various media may have a non-convex objective
function
3.1. AN ILLUSTRATIVE EXAMPLE
We illustrate the above issues with the help of the
following example. The S shaped curves in figure 1
shows the impact of various media variables on
Sales. The media variables have been labeled x, y
and z.. For example, the line labeled Sales_x depicts
the incremental sales that will result when
impressions of the media variable called x range from
0 to 205 per week. In this case, x may be weekly TV
GRPs, y may be magazine impressions per week and
z may be Online impressions per week. Notice that all
the media response curves are S shaped which
assumes that there are threshold effects associated
with each of these media variables. For the moment
we assume for simplicity that the cost per impression
is the same for each of these media variables. This
facilitates the discussion since it implies that
maximizing profits is equivalent to maximizing total
sales from a combination of these media variables.
We will also assume that each media variable is
associated with carry over effects so that each media
not only affects sales in the week in which it is aired
but also in several subsequent weeks.
The problem then resorts to finding the optimal mix of
these variables that maximize total sales.
Apart from the shape of the response curve, the
dynamic nature of marketing adds to the complexities
of the models. The effects of marketing campaigns do
not usually end when the campaign is over. At least
some of the effect of a campaign can be perceived in
future periods. In other words, sales today will be
determined not only by today's advertising but also by
advertising expenditures in previous periods. These
are called lagged effects or carryover effects of
advertising. There can be lead effects of advertising
as well. For instance, consumers sometimes
anticipate a marketing stimulus and adjust their
behavior before the stimulus actually occurs. Various
different kinds of lagged or lead effects for the media
variables are considered by marketing mix modeling
practitioners and the one that enters the model may
be a complicated transformation of the original
variable. It is not uncommon for instance, to first
create a lagged geometric series from the original
media variable and then work with logs of this
geometric series. In these situations it is very hard to
come up with an algebraic solution for the optimal
mix. The problem then resorts to one of finding the
optimal mix through an iterative process by using
Proc IML or a trial and error process by using a tool
such as the Excel solver. However, the final solutions
often depends on the initial conditions submitted and
with non linearities in the model, these methods
might end up finding a local optima rather than the
global one.
This problem, as we have defined it, has several local
optima and one global optimum as shown by the
surface chart in figure 2. The 2 horizontal axes of the
chart shows the impressions of x and y and the
vertical axis displays the total sales. We assume as
before that the cost per impression is the same for
each of these media variables and equals 1 unit per
impression. The total budget is assumed to be fixed
at 205 units so that x + y + z <= 205 must always
hold. The global optimum involves using 48 units of x,
94 units of y and 63 units of z. The corresponding
maximum sales is $10.7 million. As mentioned before
this problem also has several local optima. One local
optima involves investing in 124 units of y and 81
units of z with sales of $8.3 million. Another local
optima involves spending 68 units on x and 137 units
on y resulting in a total sales of $7.9 million. The
corner solution where we invest 205 units on y and
nothing on x and z is also a local optima since
marginal changes from this point does not lead to
increases in sales. Depending on the initial conditions
that we input in the problem, we may get any one of
the several local optima or the global one as our final
solution. With a large number of media response
variables or more non-linearities in the response
function, the number of local optima will multiply
increasing our chances of obtaining a local optimum
as the final solution instead of the global one.
Furthermore, each optimization technique under Proc
IML requires a continuous objective function and
almost all optimization subroutines (except the
NLPNMS subroutine) require continuous first order
derivatives of the objective function. It may be useful
to have an algorithm which can work even without
these restrictions. Moreover, even if a global solution
is obtained using the above methods, the solution
2
macroeconomic factors, other marketing variables,
competitor effects etc. However for simplicity we have
left those terms out of the equation and they are
assumed to remain at their original values
irrespective of the mix between the media variables.
Therefore when we refer to Sales, the reader should
keep in mind that it really means incremental sales
arising from media advertisement over and above
baseline sales that would result if the company made
no media investments whatsoever. To keep things
simple, we have assumed that there are no
interaction effects between the media variables and
any other variables in the model. In reality such
interaction effects exist and are modeled and only
add to the non-linearities and non-convexities of the
model further providing support for this analysis. Note
that while the media variables in the model are
assumed to be impressions, the analysis is similar if
media spending is used in the model instead of
media impressions. The process by which we arrive
at these variable transformations and the final model
in equation (1) – (3) is beyond the scope of this paper
and for our purpose we simply assume that the final
model has already been obtained and tested and has
been found to be satisfactory. We will use this model
to explain the procedure of obtaining the optimal
marketing mix. For simplicity we assume that we are
trying to allocate our resources between only three
media variables (Magazine, Television and Online).
We also assume that there are threshold effects
associated with each media variable so that the
relationship between each media variable and sales
is S-shaped i.e. has an initial convex and
subsequently a concave section. Several S shaped
functions can be used to depict this relationship. We
will choose a specific function called the Gompertz
function which is a non-linear S shaped function and
will be beneficial for demonstration purposes.
Assume that our data consists of weekly TV GRPs
and Magazine impressions and Online impressions
as well as product sales for about 3-4 years.
Suppose that the final model can be represented by
the following equation :
Figure 1
5
4.5
4
3.5
3
Sales_x
Sales_y
Sales_z
2.5
2
1.5
1
0.5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
110
115
120
125
130
135
140
145
150
155
160
165
170
175
180
185
190
195
200
205
0
Figure 2
$15
$10-$15
$10
$5
$5-$10
$0-$5
$0
In the following section we show how, using our
algorithm we obtain a file that displays the total sales
for each mix of media variables. The global optimum
can then be easily read of from the list. Besides, total
sales for other combinations of the media variables
can also be easily obtained. This may be useful if
during the planning stage, the media planners need
to look at various scenarios and choose a feasible
option based on different constraints.
4. A MORE UNIVERSAL METHOD:
We propose an alternative solution, using a SAS
macro, which can be used to obtain a close to global
solution for the optimal media mix. The macro uses
an iteration method to obtain the value of the
objective function for all potential allocations. The
allocation that maximizes the objective function can
then be chosen as the optimal media mix. Since the
macro works with the transformed variables, non
linear transformation and all sorts of complicated
carryover effects can be incorporated within the
macro. Furthermore, since the algorithm does not
require solving for first derivatives, the objective
function is not required to be differentiable. Below, we
outline the steps of the macro and provide a
description of how each step works.
Sales = A*b^(a^f(TV)) +
C*d^(c^g(Mag)) + E*f^(e^h(Onl))
………………………………………… (1)
where f(TV) and g(Mag) and h(Onl) are
transformations of the TV, Print and Online variables
respectively that incorporate carry over effects.
The carryover effects are assumed to be as follows.
For each media variable we assume that only a
certain fraction of the consumers who see an ad in a
particular week actually go to the store to purchase
the product in that week. The other consumers will
visit the store in one of the following weeks. We
assume that 60% of the consumers who see a TV ad
will visit the store during the week the advertisement
is aired. Therefore the transformed TV variable taking
into account carryover effects can be obtained as
follows:
f(TV) = TV60 = 0.6*TV + 0.4*lag(TV60).
Similarly, the transformed magazine and Online
variables are as follows :
g(Mag) = Mag40 = 0.4*Mag + 0.6*lag(Mag40).
h(Onl) = Onl30 = 0.3*Mag + 0.7*lag(Onl30).
We assume that the Sales equation for the product in
question is given by equation (1). Equation (1) says
that sales per week is determined by the amount of
advertisement exposures through television,
magazine and online channels. In reality we would
expect several other terms to enter the above
equation, such as influence of seasonality, holidays,
3
The paper looks at the problem of optimizing the
marketing mix subject to resource constraints. Non
linear demand curves and lagged (or lead ) effects of
advertising along with interaction effects often lead to
non convex objective functions in the marketing mix
modeling world. Standard optimization methods need
the objective functions to be convex and differentiable
and cannot be applied when these complexities exist
in the model. The algorithm outlined in this paper can
help to find the global optimum in these situations
when standard methods cannot be applied. Besides,
this algorithm helps the marketing planner to look at
various scenarios and choose the one that makes the
most business sense.
Therefore in explicit terms equation (1) can now be
written as :
Sales = A*b^(a^TV60) + C*d^(c^Mag50) +
E*f^(e^Onl30) ………………………………… (2)
We assume that the above non-linear regression
model has been solved and begets the following
coefficients for a, b, c, d, e,f, A. C and E : a=0.9,
b=0.000001, c = 0.95 , d=0.000000009 , e= 0.92, f=
0.00000001 , A=3.5 , C=4.5 , E= 4 .
Substituting these parameter values in equation(2)
gives the following model to estimate product Sales in
any week:
6. REFERENCES
Sales = 3.5*0.000001^(0.9^TV60) +
4.5*0.000000009^(0.95^Mag50) +
4*0.00000001^(0.92^Onl30) ……… (3)
Ambler, Tim. Marketing from Advertising to Zen,
Pitman Publishing, London, U.K. 1996.
4.1. HOW THE ALGORITHM WORKS:
Corkindale, David, J. Newall. Advertising thresholds
and wearout. European Journal of Marketing 12 327–
378, 1978.
Assume that during the last year, approximately 60%
of the budget was allocated to TV and 20% to
Magazine and 20% to Online. Assuming the cost per
point and flighting remains the same as last year, we
can try to see the effect on sales if the resources
were reallocated to devote say k% to TV, l% to
Magazine and (100 – k - l)% to Online. For every
value of k and l, we have a separate budget for each
media variable. If this budget is less (or greater ) than
the original budget, the algorithm prorates the media
variables to a fraction (or a multiple) of its original
value. The new values of the media variables are the
amounts that could have been purchased at the new
budget for each. We assume that the flighting
remains exactly the same as before so that the entire
media stream is prorated in exactly the same manner
in each week. For our purposes let us assume that for
k = k’ and l = l' , the new prorated TV, Magazine and
Online variables are TV’, Magazine’ and Online'
respectively. Therefore, as an example if k’ = 40%
then TV’ = (TV*40)/60. Once these new TV, Magazine
and Online variables have been obtained, the
algorithm next transforms these new variables to
obtain TV60, Mag50 and Onl30 respectively. The
sales resulting from this allocation of the budget
between TV, Magazine and Online is then easily
calculated from equation (1).
The algorithm repeats all of the above steps for all
values of k and l ( = 1, 2, ….100) and obtains the
corresponding estimated Sales. The estimated Sales
figures for different budget allocations can then be
compared and the best one which makes the most
business sense can be chosen.
While the example above explains the algorithm with
only three media variables, the macro can
accommodate additional variables. Naturally, as the
number of variables increase, the algorithm works
through a larger number of iterations which increases
the time needed for the macro to run.
Hanssens, Dominique M., Leonard J. Parsons,
Randal L. Schultz. . Market Response Models:
Econometric and Time Series Analysis ,2nd ed.
Kluwer Academic Press, Boston, M,2001.
Peter Doyle and John Saunders. Multiproduct
Advertising Budgeting, Marketing Science, Vol. 9, No.
2, (Spring 1990), pp 97-113, 1990
7. CONTACT INFORMATION
Your comments and questions are valued and
encouraged. Contact the author at:
Patralekha Bhattacharya
Thinkalytics, LLC
E-mail: [email protected]
Web: www.thinkalytics.com
SAS and all other SAS Institute Inc. product or
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Other brand and product names are trademarks of
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5. CONCLUSION
4